Invariance of plurigenera fails in positive and mixed characteristic

Abstract

We construct smooth families of elliptic surface pairs with terminal singularities over a DVR of positive or mixed characteristic $(X,B)\to \mathrm{Spec}R$, such that $P_m(X_k,B_k)>P_m(X_K,B_K)$ for all sufficiently divisible $m>0$. In particular, this shows that invariance of all sufficiently divisible plurigenera does not follow from the MMP and Abundance Conjectures.